A basketball of mass 0.70 kg is dropped from rest from a height of 1.22 m. It rebounds to a height of 0.64 m. How much mechanical energy was lost during the collision with the floor? A basketball player dribbles the ball from a height of 1.22 m by exerting a constant downward force on it for a distance of 0.13 m. In dribbling, the player compensates for the mechanical energy lost during each bounce. If the ball now returns to a height of 1.22 m, what is the magnitude of the force?
Well, I have to say that basketballs are quite bouncy, but let's see if we can put a humorous spin on this physics problem!
To find the mechanical energy lost during the collision with the floor, we need to first calculate the initial and final kinetic energies of the basketball.
The initial kinetic energy is given by the formula KE = (1/2) * m * v^2, where m is the mass of the basketball and v is the velocity. Since it is dropped from rest, the initial velocity is 0. Therefore, the initial kinetic energy is also 0.
The final velocity can be determined using the conservation of mechanical energy. We can use the formula for potential energy PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height. As the ball rebounds, it returns to a height of 0.64 m. Therefore, the final potential energy is m * g * 0.64.
Now, since the mechanical energy is conserved, the initial kinetic energy plus the initial potential energy should equal the final kinetic energy plus the final potential energy. Since the initial kinetic energy is 0, we have:
0 + (m * g * 1.22) = (1/2) * m * vf^2 + (m * g * 0.64)
Now, we can solve this equation for vf^2, which represents the final velocity squared.
vf^2 = (2 * g * (1.22 - 0.64))
Once we have the final velocity, we can calculate the final kinetic energy using the formula KE = (1/2) * m * vf^2.
Now let's move on to the second part of the question. The basketball player exerts a constant downward force to dribble the ball back to the original height of 1.22 m. The distance covered is 0.13 m.
To find the magnitude of the force, we need to calculate the work done by the player. The work done can be calculated using the formula W = F * d, where F is the force applied and d is the distance covered.
However, since the player compensates for the mechanical energy lost during each bounce, the work done by the player must be equal to the mechanical energy lost during the collision.
So, the magnitude of the force exerted by the player is equal to the mechanical energy lost (initial potential energy - final potential energy), divided by the distance covered.
I hope that helps!
To find the amount of mechanical energy lost during the collision with the floor, we need to calculate the potential energy difference before and after the bounce.
Step 1: Calculate the potential energy before the bounce.
Potential energy before the bounce (PE1) = mass (m) * gravitational acceleration (g) * height (h1)
PE1 = 0.70 kg * 9.8 m/s^2 * 1.22 m
Step 2: Calculate the potential energy after the bounce.
Potential energy after the bounce (PE2) = mass (m) * gravitational acceleration (g) * height (h2)
PE2 = 0.70 kg * 9.8 m/s^2 * 0.64 m
Step 3: Calculate the mechanical energy lost during the collision.
Mechanical energy lost = PE1 - PE2
Now let's calculate the magnitude of the force exerted by the basketball player during the dribble.
Step 4: Calculate the work done by the player.
Work done = force * distance
Work done = change in potential energy = mechanical energy lost during the bounce (from previous calculation)
Step 5: Calculate the magnitude of the force.
Force = work done / distance
Now we can use the given values in the calculations:
m = 0.70 kg
g = 9.8 m/s^2
h1 = 1.22 m
h2 = 0.64 m
distance = 0.13 m
Calculating the mechanical energy lost:
PE1 = 0.70 kg * 9.8 m/s^2 * 1.22 m
PE2 = 0.70 kg * 9.8 m/s^2 * 0.64 m
Mechanical energy lost = PE1 - PE2
Calculating the magnitude of the force:
Force = (Mechanical energy lost) / distance
To find the amount of mechanical energy lost during the collision with the floor, we need to calculate the difference in mechanical energy before and after the bounce.
The initial mechanical energy (E_initial) of the basketball can be calculated using the formula:
E_initial = m*g*h_initial
where:
m = mass of the basketball (0.70 kg)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h_initial = initial height (1.22 m)
E_initial = (0.70 kg) * (9.8 m/s^2) * (1.22 m)
E_initial = 8.2956 Joules (rounded to four decimal places)
The final mechanical energy (E_final) can be calculated using the height after the bounce (h_final) and the same formula:
E_final = m*g*h_final
where:
h_final = final height after the bounce (0.64 m)
E_final = (0.70 kg) * (9.8 m/s^2) * (0.64 m)
E_final = 4.34048 Joules (rounded to five decimal places)
The amount of mechanical energy lost (ΔE) can be obtained by subtracting E_final from E_initial:
ΔE = E_initial - E_final
ΔE = 8.2956 Joules - 4.34048 Joules
ΔE = 3.95512 Joules (rounded to five decimal places)
Now, to find the magnitude of the force exerted by the basketball player during dribbling, we can use the work-energy principle:
Work done on the ball = change in mechanical energy
Since the work done on the ball is equal to the force (F) multiplied by the distance (d) over which the force is applied, we can write:
F * d = ΔE
Rearranging the equation, we can solve for the magnitude of the force (F):
F = ΔE / d
where:
ΔE = amount of mechanical energy lost during the bounce (3.95512 Joules, as calculated earlier)
d = distance over which the force is applied (0.13 m)
F = (3.95512 Joules) / (0.13 m)
F = 30.504 Joules/m
Therefore, the magnitude of the force exerted by the basketball player during dribbling is approximately 30.504 Joules per meter.